Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912937708355584 |
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| author | Cho, Soojin Huh, JiSun Park, Seonjeong |
| author_facet | Cho, Soojin Huh, JiSun Park, Seonjeong |
| contents | A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell intersected with the Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type $A$ in this paper.
We consider the smoothness of the intersection of a Schubert variety and a Hessenberg variety to ensure the smoothness of the corresponding Hessenberg Schubert variety. Specifically, we analyze the structure of the GKM graphs of the intersection of a Schubert variety and a Hessenberg variety. Our results show that the regularity of these GKM graphs is completely characterized in terms of pattern avoidance, which is a necessary and sufficient condition for the intersection to be smooth. This shows that our pattern avoidance provides a sufficient condition for the smoothness of a Hessenberg Schubert variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_13334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties Cho, Soojin Huh, JiSun Park, Seonjeong Combinatorics Algebraic Geometry 05E14, 14M15 A \emph{Hessenberg Schubert variety} is an irreducible component of the intersection of a Schubert variety and a Hessenberg variety, defined as the closure of a Schubert cell intersected with the Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type $A$ in this paper. We consider the smoothness of the intersection of a Schubert variety and a Hessenberg variety to ensure the smoothness of the corresponding Hessenberg Schubert variety. Specifically, we analyze the structure of the GKM graphs of the intersection of a Schubert variety and a Hessenberg variety. Our results show that the regularity of these GKM graphs is completely characterized in terms of pattern avoidance, which is a necessary and sufficient condition for the intersection to be smooth. This shows that our pattern avoidance provides a sufficient condition for the smoothness of a Hessenberg Schubert variety. |
| title | Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties |
| topic | Combinatorics Algebraic Geometry 05E14, 14M15 |
| url | https://arxiv.org/abs/2307.13334 |